Chapter 15, Problem 8PS

A

Summary Introduction

To calculate: The expected price of the 4-year bond at the end of the first year, second year, third year and fourth year are to be determined.

Introduction: When the forward rates are equal to the market expectation rates is called as expectation hypothesis. The expected rate of return is defined as the amount which is expected on a security at specific period.

Answer to Problem 8PS

The expected price of the 4-year bond is shown as −

Beginning of the year	Expected price
1	$792.16
2	$839.69
3	$881.68
4	$934.58

Explanation of Solution

Expectations theory is the long term interest rate that predicts the short term interest rates. It suggests the investor gets same interest by investing in two different investment having diffrent maturity period.At this condition liquidity premium is zero.

The following method will be used for the calculation of the Yield to maturity (YTM) and the forward rate −

  $YTM={\left(\frac{FaceValue}{CurrentValue}\right)}^{\frac{1}{YearsToMaturity}}$ …......................Equ (1)

  $ForwardRat{e}_{CurrentYear}=\frac{\left(1+YT{M}_{CurrentYear}\right)}{\left(1+YT{M}_{\mathrm{Pr}eviousYear}\right)}-1$...................... Equ (2)

Maturity	Price of bond	YTM	Forward rate
1	$943.40	$\left[{\left(\frac{1000}{943.4}\right)}^{(1/1)}\right]-1$
2	$898.47	$\left[{\left(\frac{1000}{898.47}\right)}^{(1/2)}\right]-1$	$[{(1+5.499\%)}^2/{(1+6\%)}^1]-1$
3	$847.62	$\left[{\left(\frac{1000}{847.62}\right)}^{(1/3)}\right]-1$	$[{(1+5.665\%)}^3/{(1+5.499\%)}^2]-1$
4	$792.16	$\left[{\left(\frac{1000}{792.16}\right)}^{(1/4)}\right]-1$	$[{(1+5.998\%)}^4/{(1+5.665\%)}^3]-1$

On calculation, the values of forward rate and YTM is given as −

Maturity	Price of bond	YTM	Forward rate
1	$943.40	6.00%	6.00%
2	$898.47	5.50%	5.00%
3	$847.62	5.67%	6.00%
4	$792.16	6.00%	7.00%

Now, the following method will be used for the calculation of the expected price −

Beginning of the year	Expected price calculation	Expected price
1	$792.16	$792.16
2	$\$1000/\left(\left(1+5\%\right)\times \left(1+6\%\right)\times \left(1+7\%\right)\right)$	$839.69
3	$\$1000/\left(\left(1+6\%\right)\times \left(1+7\%\right)\right)$	$881.68
4	$\$1000/\left(\left(1+7\%\right)\right)$	$934.58

The expected price of the 4-year bond is given as −

Beginning of the year	Expected price
1	$792.16
2	$839.69
3	$881.68
4	$934.58

B

Summary Introduction

To calculate: The rate of return of the bond in first year, second year, third year and fourth year and prove that expected return equals the forward rate for each year.

Introduction: When the forward rates are equal to the market expectation rates is called as expectation hypothesis. The expected rate of return is defined as the amount which is expected on a security at specific period.

Answer to Problem 8PS

The forward rate and expected rate of return is equal.

Explanation of Solution

Expectations theory is the long term interest rate that predicts the short term interest rates. It suggests the investor gets same interest by investing in two different investment having diffrent maturity period. At this condition liquidity premium is zero.

The following formula will be used for the calculation of the return of year bond −

Beginning of the year	Expected price	Expected rate of return calculation	Expected rate of return
1	$792.16	$(\$839.69/\$792.16)-1$	6.00%
2	$839.69	$(\$881.68/\$839.69)-1$	5.00%
3	$881.68	$(\$934.58/\$881.68)-1$	6.00%
4	$934.58	$(\$1000/\$934.58)-1$	7.00%

Now, the comparison between the values of the forward rate and the expected rate of return is given as −

Forward rate	Expected rate of return
6.00%	6.00%
5.00%	5.00%
6.00%	6.00%
7.00%	7.00%

The above table proves that the value of the forward rate is equal to the value of the expected rate of return for each year.